Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [552,2,Mod(5,552)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(552, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 11, 11, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("552.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 552 = 2^{3} \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 552.bf (of order \(22\), degree \(10\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(4.40774219157\) |
Analytic rank: | \(0\) |
Dimension: | \(920\) |
Relative dimension: | \(92\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.41419 | + | 0.00734365i | 0.191591 | − | 1.72142i | 1.99989 | − | 0.0207707i | 3.15950 | + | 0.454268i | −0.258305 | + | 2.43583i | 0.966873 | − | 0.441556i | −2.82808 | + | 0.0440603i | −2.92659 | − | 0.659617i | −4.47149 | − | 0.619221i |
5.2 | −1.41317 | + | 0.0543574i | 0.873168 | − | 1.49585i | 1.99409 | − | 0.153632i | −1.97986 | − | 0.284661i | −1.15262 | + | 2.16136i | 2.90361 | − | 1.32603i | −2.80963 | + | 0.325502i | −1.47515 | − | 2.61226i | 2.81335 | + | 0.294653i |
5.3 | −1.41053 | − | 0.101993i | −1.59542 | − | 0.674260i | 1.97919 | + | 0.287730i | 3.02191 | + | 0.434486i | 2.18162 | + | 1.11379i | −3.87309 | + | 1.76878i | −2.76237 | − | 0.607716i | 2.09075 | + | 2.15146i | −4.21819 | − | 0.921070i |
5.4 | −1.40980 | − | 0.111635i | −1.53275 | + | 0.806651i | 1.97508 | + | 0.314766i | −0.920190 | − | 0.132303i | 2.25092 | − | 0.966109i | −0.322471 | + | 0.147268i | −2.74932 | − | 0.664245i | 1.69863 | − | 2.47278i | 1.28251 | + | 0.289247i |
5.5 | −1.40646 | − | 0.147850i | 1.72844 | − | 0.111718i | 1.95628 | + | 0.415891i | −2.37700 | − | 0.341761i | −2.44751 | − | 0.0984225i | −3.17635 | + | 1.45059i | −2.68995 | − | 0.874172i | 2.97504 | − | 0.386198i | 3.29264 | + | 0.832114i |
5.6 | −1.39862 | − | 0.209415i | 1.28405 | + | 1.16242i | 1.91229 | + | 0.585785i | 1.33695 | + | 0.192224i | −1.55247 | − | 1.89469i | 3.52408 | − | 1.60939i | −2.55190 | − | 1.21976i | 0.297559 | + | 2.98521i | −1.82963 | − | 0.548825i |
5.7 | −1.36422 | + | 0.372693i | −1.31009 | − | 1.13298i | 1.72220 | − | 1.01687i | −2.59161 | − | 0.372618i | 2.20951 | + | 1.05738i | 0.360822 | − | 0.164782i | −1.97048 | + | 2.02909i | 0.432691 | + | 2.96863i | 3.67440 | − | 0.457543i |
5.8 | −1.35818 | + | 0.394141i | 0.367339 | + | 1.69265i | 1.68931 | − | 1.07063i | −0.449826 | − | 0.0646752i | −1.16605 | − | 2.15414i | −0.674934 | + | 0.308232i | −1.87240 | + | 2.11993i | −2.73012 | + | 1.24355i | 0.636436 | − | 0.0894541i |
5.9 | −1.34520 | + | 0.436397i | 1.59574 | + | 0.673497i | 1.61912 | − | 1.17408i | 1.00923 | + | 0.145106i | −2.44050 | − | 0.209609i | 0.340247 | − | 0.155386i | −1.66567 | + | 2.28595i | 2.09280 | + | 2.14946i | −1.42094 | + | 0.245230i |
5.10 | −1.32050 | + | 0.506245i | −0.661683 | + | 1.60068i | 1.48743 | − | 1.33699i | 3.61745 | + | 0.520111i | 0.0634149 | − | 2.44867i | −0.429427 | + | 0.196113i | −1.28731 | + | 2.51850i | −2.12435 | − | 2.11829i | −5.04014 | + | 1.14451i |
5.11 | −1.28981 | − | 0.579981i | −1.33333 | − | 1.10555i | 1.32724 | + | 1.49614i | 1.33695 | + | 0.192224i | 1.07855 | + | 2.19926i | 3.52408 | − | 1.60939i | −0.844167 | − | 2.69952i | 0.555524 | + | 2.94812i | −1.61293 | − | 1.02334i |
5.12 | −1.26748 | + | 0.627300i | −1.72666 | + | 0.136498i | 1.21299 | − | 1.59018i | 1.70656 | + | 0.245367i | 2.10288 | − | 1.25614i | 4.42418 | − | 2.02045i | −0.539916 | + | 2.77642i | 2.96274 | − | 0.471373i | −2.31695 | + | 0.759532i |
5.13 | −1.26313 | − | 0.636012i | −0.135402 | − | 1.72675i | 1.19098 | + | 1.60673i | −2.37700 | − | 0.341761i | −0.927205 | + | 2.26722i | −3.17635 | + | 1.45059i | −0.482454 | − | 2.78698i | −2.96333 | + | 0.467611i | 2.78509 | + | 1.94349i |
5.14 | −1.26067 | + | 0.640864i | 0.0422748 | − | 1.73153i | 1.17859 | − | 1.61584i | 0.349508 | + | 0.0502517i | 1.05638 | + | 2.20999i | −2.03405 | + | 0.928918i | −0.450281 | + | 2.79236i | −2.99643 | − | 0.146400i | −0.472820 | + | 0.160636i |
5.15 | −1.24635 | − | 0.668282i | −0.580308 | + | 1.63194i | 1.10680 | + | 1.66583i | −0.920190 | − | 0.132303i | 1.81387 | − | 1.64617i | −0.322471 | + | 0.147268i | −0.266214 | − | 2.81587i | −2.32649 | − | 1.89406i | 1.05847 | + | 0.779843i |
5.16 | −1.24176 | − | 0.676788i | 0.894449 | + | 1.48323i | 1.08392 | + | 1.68081i | 3.02191 | + | 0.434486i | −0.106857 | − | 2.44716i | −3.87309 | + | 1.76878i | −0.208404 | − | 2.82074i | −1.39992 | + | 2.65334i | −3.45842 | − | 2.58472i |
5.17 | −1.18573 | − | 0.770749i | 1.67663 | − | 0.434625i | 0.811892 | + | 1.82779i | 3.15950 | + | 0.454268i | −2.32301 | − | 0.776919i | 0.966873 | − | 0.441556i | 0.446090 | − | 2.79303i | 2.62220 | − | 1.45741i | −3.39618 | − | 2.97382i |
5.18 | −1.15945 | − | 0.809745i | 1.35636 | − | 1.07716i | 0.688626 | + | 1.87771i | −1.97986 | − | 0.284661i | −2.44486 | + | 0.150603i | 2.90361 | − | 1.32603i | 0.722042 | − | 2.73471i | 0.679440 | − | 2.92205i | 2.06503 | + | 1.93323i |
5.19 | −1.13324 | + | 0.846027i | −1.52607 | + | 0.819214i | 0.568475 | − | 1.91751i | −4.20271 | − | 0.604259i | 1.03633 | − | 2.21946i | −3.19599 | + | 1.45956i | 0.978044 | + | 2.65395i | 1.65778 | − | 2.50036i | 5.27391 | − | 2.87084i |
5.20 | −1.09564 | + | 0.894186i | 1.69680 | + | 0.347656i | 0.400861 | − | 1.95942i | −3.74972 | − | 0.539128i | −2.16996 | + | 1.13635i | 1.49121 | − | 0.681011i | 1.31288 | + | 2.50526i | 2.75827 | + | 1.17981i | 4.59043 | − | 2.76226i |
See next 80 embeddings (of 920 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
24.h | odd | 2 | 1 | inner |
69.g | even | 22 | 1 | inner |
184.m | odd | 22 | 1 | inner |
552.bf | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 552.2.bf.a | ✓ | 920 |
3.b | odd | 2 | 1 | inner | 552.2.bf.a | ✓ | 920 |
8.b | even | 2 | 1 | inner | 552.2.bf.a | ✓ | 920 |
23.d | odd | 22 | 1 | inner | 552.2.bf.a | ✓ | 920 |
24.h | odd | 2 | 1 | inner | 552.2.bf.a | ✓ | 920 |
69.g | even | 22 | 1 | inner | 552.2.bf.a | ✓ | 920 |
184.m | odd | 22 | 1 | inner | 552.2.bf.a | ✓ | 920 |
552.bf | even | 22 | 1 | inner | 552.2.bf.a | ✓ | 920 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
552.2.bf.a | ✓ | 920 | 1.a | even | 1 | 1 | trivial |
552.2.bf.a | ✓ | 920 | 3.b | odd | 2 | 1 | inner |
552.2.bf.a | ✓ | 920 | 8.b | even | 2 | 1 | inner |
552.2.bf.a | ✓ | 920 | 23.d | odd | 22 | 1 | inner |
552.2.bf.a | ✓ | 920 | 24.h | odd | 2 | 1 | inner |
552.2.bf.a | ✓ | 920 | 69.g | even | 22 | 1 | inner |
552.2.bf.a | ✓ | 920 | 184.m | odd | 22 | 1 | inner |
552.2.bf.a | ✓ | 920 | 552.bf | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(552, [\chi])\).